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Given parameters $\lambda, \nu>0$, a covariance matrix $R$, a mean vector $\mu \in R^p$, the Arellano-Valle and Bolfarine's generalized $t$ distribution is given by (see, for example, the book by Kotz and Nadarajah) \begin{equation} f(x)=\frac{\Gamma((\nu+p)/2)}{(\pi)^\frac{p}{2}\Gamma(\nu/2)|R|^\frac{1}{2}} \left[ 1+\frac{1}{\lambda}(x-\mu)^T R^{-1} (x-\mu) \right]^{-(\nu+p)/2}, \end{equation} where $x \in R^p$.

I want to find the expectation, under $f(x)$ of something similar to the density, i.e., I want to evaluate

\begin{equation} \int_x f(x) \left[ 1+\frac{1}{\lambda_2}(x-\mu_2)^T R_2^{-1} (x-\mu_2) \right]^{-\nu_2} dx, \end{equation} where $R_2$ is a covariance matrix, $\mu_2$ is a vector, and $\lambda_2>0$.

Is there a closed form expression for this integral for $x \in R^p$? In 1-d?

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up vote 0 down vote accepted

I am not sure a closed-form expression is known in the literature, but I found this paper that describes other properties of that integral without solving it.

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